Changes between Version 5 and Version 6 of u/erica/scratch3


Ignore:
Timestamp:
04/03/16 14:11:46 (9 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/scratch3

    v5 v6  
    3030== Optically thick limit ==
    3131
    32 And we see that as [[latex($R\rightarrow 0$)]], [[latex($\lambda\rightarrow \frac{1}{3}$)]]. How to interpret this? R is essentially the ratio of the optical depth [[latex($\tau=1/\kappa$)]], to the radiation's scale height, [[latex($L^{-1}=\frac{\nabla E}{E}$)]].
     32From the graph above, we see that as [[latex($R\rightarrow 0$)]], [[latex($\lambda\rightarrow \frac{1}{3}$)]]. How to interpret this? R is essentially the ratio of the optical depth [[latex($\tau=1/\kappa$)]], to the radiation's scale height, [[latex($L^{-1}=\frac{\nabla E}{E}$)]]. Thus, as [[latex($R\rightarrow 0$)]], we have:
    3333
    34 Thus, as [[latex($R\rightarrow 0$)]], we have that [[latex($\frac{\tau}{L}\rightarrow 0$)]] - i.e. radiation travels infinitesmally small distances before it is absorbed or scattered - and thus, we are in the optically thick regime.
     34[[latex($\boxed{\frac{\tau}{L}\rightarrow 0 ~,~ \lambda\rightarrow \frac{1}{3}}$)]]
     35
     36That is, the radiation travels infinitesmally small distances before it is absorbed or scattered - and thus, we are in the optically thick regime.
    3537
    3638The rad diffusion equation then becomes,
     
    3840[[latex($\frac{\partial E}{\partial t} = \nabla \cdot (\frac{1}{3}\frac{c}{\kappa_R \rho} \nabla E)$)]]
    3941
    40 which is equation 6.59 in Drake's book.
     42which is consistent with equation 6.59 in Drake's book describing optically thick, non-equilibrium radiation transfer. 
    4143
    4244== Free streaming limit ==
     45
     46In the other limit, [[latex($R\rightarrow \infty$)]], we have:
     47
     48[[latex($\boxed{\frac{\tau}{L}\rightarrow \infty~,~\lambda\rightarrow 0}$)]]
     49
     50That is, a photon travels an infinite distance before it interacts with another particle -- i.e. we are in the free-streaming limit.